Properties

Label 714f
Number of curves $4$
Conductor $714$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("714.e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 714f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
714.e4 714f1 [1, 1, 1, 1, 101] [4] 192 \(\Gamma_0(N)\)-optimal
714.e3 714f2 [1, 1, 1, -319, 2021] [2, 2] 384  
714.e2 714f3 [1, 1, 1, -679, -3883] [2] 768  
714.e1 714f4 [1, 1, 1, -5079, 137205] [2] 768  

Rank

sage: E.rank()
 

The elliptic curves in class 714f have rank \(1\).

Modular form 714.2.a.e

sage: E.q_eigenform(10)
 
\( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2q^{10} - q^{12} - 6q^{13} - q^{14} + 2q^{15} + q^{16} + q^{17} + q^{18} + O(q^{20}) \)

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with Cremona labels.